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Streak evolution in viscoelastic Couette flow

Published online by Cambridge University Press:  21 February 2014

Jacob Page  and
Tamer A. Zaki
Jacob Page
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
Tamer A. Zaki
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
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Abstract

The combined effect of inertia and elasticity on streak amplification in planar Couette flow of an Oldroyd-B fluid is examined. The linear perturbation equations are solved in the form of a forced-response problem to obtain the wall-normal vorticity response to a decaying streamwise vortex. With significant disparity between the solvent diffusion and polymer relaxation time scales, two distinct responses are possible. The first is termed ‘quasi-Newtonian’ because the streak evolution collapses onto the Newtonian behaviour at the same total and solvent Reynolds numbers when relaxation is very fast or slow, respectively. The second response is labelled ‘elastic’: with a long relaxation time, the streaks can reach significant amplitudes even with very weak inertia. If the diffusion and relaxation time scales are commensurate, the streaks are able to re-energize in a periodic cycle within an envelope of overall decay. This behaviour is enhanced in the instantaneously elastic limit, where the governing equation reduces to a forced wave equation. The streak re-energization is demonstrated to be a superposition of trapped vorticity waves.

JFM classification

Instability: Transition to turbulence Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 520 - 551
Copyright
© 2014 Cambridge University Press 

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References

Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.CrossRefGoogle Scholar
Azaiez, J. & Homsy, G. M. 1994 Linear stability of free shear flow of viscoelastic liquids. J. Fluid Mech. 268, 37–69.CrossRefGoogle Scholar
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 3258.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, vol.1. 2nd edn. Wiley.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Denn, M. & Porteous, K. 1971 Elastic effects in flow of viscoelastic liquids. Chem. Engng J. 2, 280–286.CrossRefGoogle Scholar
Doering, C. R., Eckhardt, B. & Schumacher, J. 2006 Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers. J. Non-Newtonian Fluid Mech. 135, 92–96.CrossRefGoogle Scholar
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25, 110817.CrossRefGoogle ScholarPubMed
Dubief, Y., White, C. M., Terrapon, V. E., Shaqfeh, E. S. G., Moin, P. & Lele, S. K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280.CrossRefGoogle Scholar
Graham, M. D. 1999 The sharkskin instability of polymer melt flows. Chaos 9, 154–163.CrossRefGoogle ScholarPubMed
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.Google Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407–424.CrossRefGoogle Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2009 Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. Fluid Mech. 625, 411–434.CrossRefGoogle Scholar
James, D. F. 2009 Boger fluids. Annu. Rev. Fluid Mech. 41, 129–142.CrossRefGoogle Scholar
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.CrossRefGoogle Scholar
Joseph, D. D., Narain, A. & Riccius, O. 1986 Shear-wave speeds and elastic moduli for different liquids. Part 1. Theory. J. Fluid Mech. 171, 289–308.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Jovanovic, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22, 023101.CrossRefGoogle Scholar
Jovanovic, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166, 755–778.CrossRefGoogle Scholar
Kupferman, R. 2005 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation. J. Non-Newtonian Fluid Mech. 127, 169–190.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Larson, R. G. 2000 Turbulence without inertia. Nature 405, 27–28.CrossRefGoogle ScholarPubMed
Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely elastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573–600.CrossRefGoogle Scholar
Lieu, B. K., Jovanović, M. R. & Kumar, S. 2013 Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids. J. Fluid Mech. 723, 232–263.CrossRefGoogle Scholar
Min, T., Yul Yoo, J., Choi, H. & Joseph, D. 2003 Drag reduction by polymer additives in a turbulent channel flow. J. Fluid Mech. 486, 213–238.CrossRefGoogle Scholar
Morozov, A. N. & Saarloos, W. V. 2007 An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep. 447, 112–143.CrossRefGoogle Scholar
Nouar, C., Bottaro, A. & Brancher, J. P. 2007 Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids. J. Fluid Mech. 592, 177–194.CrossRefGoogle Scholar
Pan, L., Morozov, A., Wagner, C. & Arratia, P. E. 2013 Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110, 174502.CrossRefGoogle ScholarPubMed
Preziosi, L. & Joseph, D. D. 1987 Stokes’ first problem for viscoelastic fluids. J. Non-Newtonian Fluid Mech. 25, 239–259.CrossRefGoogle Scholar
Rallison, J. M. & Hinch, E. J. 1995 Instability of a high-speed submerged elastic jet. J. Fluid Mech. 288, 311–324.CrossRefGoogle Scholar
Ray, P. K. & Zaki, T. A. 2014 Absolute instability in viscoelastic mixing layers. Phys. Fluids 26, 014103.CrossRefGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Renardy, M. 2009 Stress modes in linear stability of viscoelastic flows. J. Non-Newtonian Fluid Mech. 159, 137–140.CrossRefGoogle Scholar
Samanta, D. S., Dubief, Y., Holzner, H., Schäfer, C., Morozov, A. N., Wagner, C. & Hof, B. 2013 Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 110, 10557–10562.CrossRefGoogle ScholarPubMed
Schmid, P. J. 2007 Nonmodal Stability Theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.CrossRefGoogle Scholar
Tanner, R. I. 1962 Note on the Rayleigh problem for a visco-elastic fluid. Z. Angew. Math. Phys. 13, 573–580.CrossRefGoogle Scholar
Terrapon, V. E., Dubief, Y., Moin, P., Shaqfeh, E. S. G. & Lele, S. K. 2004 Simulated polymer stretch in a turbulent flow using Brownian dynamics. J. Fluid Mech. 504, 61–71.CrossRefGoogle Scholar
Vaughan, N. J. & Zaki, T. A. 2011 Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks. J. Fluid Mech. 681, 116–153.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.CrossRefGoogle Scholar
Zhang, M., Lashgari, I., Zaki, T. A. & Brandt, L. 2013 Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech. 737, 249–279.CrossRefGoogle Scholar
Zhou, L., Cook, L. P. & McKinley, G. H. 2012 Multiple shear-banding transitions for a model of wormlike micellar solutions. SIAM J. Appl. Math. 72, 1192–1212.CrossRefGoogle Scholar